Magnetohydrodynamic (MHD) turbulence is a crucial component of the current paradigms of star formation, dynamo theory, particle transport, magnetic reconnection and evolution of structure in the interstellar medium (ISM) of galaxies. Despite the importance of turbulence to astrophysical fluids, a full theoretical framework based on solutions to the Navier-Stokes equations remains intractable. Observations provide only limited line-of-sight information on densities, temperatures, velocities and magnetic field strengths and therefore directly measuring turbulence in the ISM is challenging. A statistical approach has been of great utility in allowing comparisons of observations, simulations and analytic predictions. In this review article we address the growing importance of MHD turbulence in many fields of astrophysics and review statistical diagnostics for studying interstellar and interplanetary turbulence. In particular, we will review statistical diagnostics and machine learning algorithms that have been developed for observational data sets in order to obtain information about the turbulence cascade, fluid compressibility (sonic Mach number), and magnetization of fluid (Alfvenic Mach number). These techniques have often been tested on numerical simulations of MHD turbulence, which may include the creation of synthetic observations, and are often formulated on theoretical expectations for compressible magnetized turbulence. We stress the use of multiple techniques, as this can provide a more accurate indication of the turbulence parameters of interest. We conclude by describing several open-source tools for the astrophysical community to use when dealing with turbulence.
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